# How to use CDF and PDF statistics for analysis

This may be too much of a general question but I hope I can find help here. I am starting a RA job in my university and my topic will be related to Internet Traffic Analysis. I am fairly new to the world of analysis but I guess in the world of research this is what I have to do a lot of.

I've been through a few papers and in a lot of them I find they use Probability Density (PDF), CDF, CCDF etc to explain the results they've obtained. For example, PDF of user session duration, CDF of bytes transferred each day etc. I have taken probability and statistics class so I understand what they are but I am still confused to the cases in which such representation would be chosen.

So, if there is anyone out there who does such graphs and analysis (in any other topic in general or this) could you tell me simply in what situation would I use one or the other of these representations

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It's partly a matter of taste and convention, but theory, attention to your objectives, and a smidgen of cognitive neuroscience [see the references] can provide some guidance.

Because a pdf and a cdf convey the same information, the distinction between them arises from how they do it: a pdf represents probability with areas while a cdf represents probability with (vertical) distances. Studies show that people compare distances faster and more accurately than they compare areas and that they systematically mis-estimate areas. Thus, if your purpose is to provide a graphical tool for reading off probabilities, your should favor using a cdf.

Pdfs and cdfs also represent probability density: the former does so by means of height while the latter represents density by slope. Now the tables are turned, because people are poor estimators of slope (which is the tangent of an angle; we tend to see the angle itself). Densities are good at conveying information about modes, heaviness of tails, and gaps. Favor using pdfs in such situations and anywhere else where local details of the probability distribution need to be emphasized.

Sometimes a pdf or cdf provides useful theoretical information. Its value (or rather the inverse thereof) is involved in formulas for standard errors for quantiles, extremes, and rank statistics. Display a pdf rather than a cdf in such situations. When studying multivariate correlations in a nonparametric setting, such as with copulas, the cdf turns out to be more useful (perhaps because it is the function that transforms a continuous probability law into a uniform one).

A pdf or cdf can be intimately associated with a particular statistical test. The Kolmogorov-Smirnov test (and the KS statistic) has a simple graphical representation in terms of a vertical buffer around the cdf; it has no simple graphical representation in terms of the pdf (that I know of).

The ccdf (complementary cdf) is used in special applications that focus on survivorship and rare events. Its use tends to be established by convention.

## References

W.S. Cleveland (1994). The Elements of Graphing Data. Summit, NJ, USA: Hobart Press. ISBN 0-9634884-1-4

B.D. Dent (1999). Cartography: Thematic Map Design 5th Ed. Boston, MA, USA: WCB McGraw-Hill.

A.M. MacEachren (2004). How Maps Work. New York, NY, USA: The Guilford Press. ISBN 1-57230-040-X

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(+1) especially for the insights in the interpretability of distances vs areas and slope vs height. – steffen Nov 24 '10 at 16:35

The CDF has a simple non-parametric estimator that needs no choices to be made: the empirical distribution function. It's not quite so simple to estimate a PDF. If you use a histogram you need to choose the bin width and the starting point for the first bin. If you use kernel density estimation you need to choose the kernel shape and bandwidth. A suspicious or cynical reader may wonder if you really chose these entirely a priori or if you tried a few different values and chose the ones that gave the result you most liked.

This is only a minor point though. The ones whuber made are more important, so i'd probably only use this to choose when I was still undecided after considering those.

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It's still an interesting point. Thanks for bringing it up. – whuber Nov 24 '10 at 19:40

I guess it depends on what statistics or findings you are going to find out, research, study, or report. I'm assuming you will prob be using these graphs to represent findings for your university topic, right?

Like for example, if you want to present your finding about say, 'How long users stays on a a certain website', it may be good to show it in CDF as it shows the accumulated time he spent on that website, through the pages etc.

On the other hand, if you want to simply show the probability of users clicking on an advert link (e.g. Google adwords link) then you may want to present it in PDF form as it will probably be a normal distribution bell curve and you can show the probability of that heppening.

Hope this helps, Jeff

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